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http://dx.doi.org/10.4134/JKMS.j190646

BIFURCATION PROBLEM FOR A CLASS OF QUASILINEAR FRACTIONAL SCHRÖDINGER EQUATIONS  

Abid, Imed (University of Tunis El Manar Higher Institute of Medical Technologies of Tunis)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.6, 2020 , pp. 1347-1372 More about this Journal
Abstract
We study bifurcation for the following fractional Schrödinger equation $$\{\left.\begin{eqnarray}(-{\Delta})^su+V(x)u&=&{\lambda}f(u)&&{\text{in}}\;{\Omega}\\u&>&0&&{\text{in}}\;{\Omega}\\u&=&0&&{\hspace{32}}{\text{in}}\;{\mathbb{R}}^n{\backslash}{\Omega}\end{eqnarray}\right$$ where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of ℝn, (-∆)s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is $\lim_{t{\rightarrow}+{\infty}}\;{\frac{f(t)}{t}}=a{\in}(0,+{\infty})$. We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.
Keywords
Bifurcation problems; fractional partial differential equations; fractional $Schr{\ddot{o}}dinger$ equations; weak solution; stability;
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